The Unwavering Mask
As you probably have gleaned from Byakuya's Aesthetic, he is generally a very reserved, lofty man that would tremendously benefit from some laughter in his life. Now let's look closer at some of the intricacies Kubo has painted into his personality, though the majority will be explored in the other pages of the Personal and Relational sections.
Calculated Arrogance
With that level of strength, did you really think you could break my blade?
When challenged by someone beneath himself (read: 95% of the Bleach population), Byakuya typically responds with either demeaning, flatly delivered insults or flowery variations of "I'm better than you, so hop off." Not very admirable things to do, but you can't blame him for feeling entitled. He was raised in one of the Four Noble Houses as the heir to clan leadership and the 6th Division captainship. He has since maintained the Kuchiki reputation in spite of personal indiscretions and proven himself to be one of the most capable shinigami currently active in Soul Society. So yeah, that ego is pretty darn justified.
Since Byakuya serves as the primary antagonist in the Soul Society arc, this quality really gives the protagonists something to fight against. Thereafter, it's just cool to see his confidence rewarded when he fells an enemy using as little perceived effort as possible. It helps balance the dramatic, uncertain clashes in the series.
Dutiful
We have to set an example for all shinigami. If we do not uphold the law, who will?
Byakuya's commitment to the law, his clan, and his shinigami responsibilities trump his self-importance any day of the week. In his view, his talents and status do not place him above the system, but rather compel him to safeguard it. Chaos will likely never encounter a more unyielding enemy.
While this sense of duty poisons him into alienating his adopted sister Rukia, it later becomes the source that enables him to protect her. He may have been driven enough to protect the law at the cost of his personal wishes, but he has also shown in times past and future that he can be just as committed to his own desires to protect what is precious to him.
As described, this steadfastness is Byakuya's greatest source of conflict, his most inhumane quality, and also his most admirable. Everyone struggles with making choices and sticking to them because of factors often outside of our control. For this captain, it seems that even if he has to tear his heart out in the process, his course of action is absolute once charted.
Quirky(?)
When you have someone as uptight as Byakuya, it's a given that his demeanor can and will be used against him. Enter Wakame Taishi, or the "Seaweed Ambassador." It pops up a few times in both manga and anime extras as Byakuya's go-to...original character(?) for items like pastries, sand sculptures, and cookies. He is, of course, completely oblivious to the fact that everyone but Rukia (who has her own bizarre art style) is either dumbfounded or grossed out by it.
While we never see this type of silliness in the regular flow of the story, it's refreshing to see that Byakuya has at least a quirk or two to curb his seriousness. I might even venture to say that humor involving him tends to be some of the funniest, because it's so unexpected that even the smallest gag is ridiculous enough to laugh at. Or maybe that's just bias talking...you decide!